ON THE GENERALIZATION OF SOME CLASSES OF CONVEX IN DIRECTION AND TYPICALLY REAL FUNCTIONS

In the article by M.O. Reade (Duke Math. Journal, 1956) using the condition |arg (f'(z)/ g'(z)) | ≤ γπ/2, where g(z) is a convex function,0≤γ≤1 , a class of functions close-to-convex (almost convex) of order γ is introduced. In our paper, we introduce a subclass of the class of close-to-convex (almost convex) order γ functions satisfying the condition |arg [(1-λzn ) f'(z)]| ≤ γπ/2, which, for different parameter values, gives a number of well-known subclasses of univalent (schlicht) functions. Based on this subclass, a class of close-to-starlike (almost star-shaped) functions is constructed, containing a number of subclasses that have been actively studied by many authors in recent years, as well as a classical class of typically real functions. For this classes exact theorems of distortion (growth) and radii of convexity (starlikeness) are obtained, generalizing previously known results. The case is also considered when the functions of the introduced classes have missing members in the power series expansion. The results obtained are accurate and not only generalize previously known results, but also reveal the properties of a number of new subclasses of univalent (schlicht) functions.